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Evaluate each expression without using a calculator. log2 64
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 5, Problema 25
In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. ln e5
Guida verificata passo dopo passo1
Recall the property of logarithms that states: \(\ln\left(e^x\right) = x\). This is because the natural logarithm function \(\ln(x)\) is the inverse of the exponential function \(e^x\).
Identify the expression inside the logarithm: \(e^5\). Here, the exponent is 5.
Apply the property directly to simplify the expression: \(\ln\left(e^5\right) = 5\).
Since the logarithm and the exponential functions are inverses, the expression simplifies exactly to the exponent without any further calculation.
Therefore, the value of \(\ln\left(e^5\right)\) is simply 5.
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Natural Logarithm (ln)
The natural logarithm, denoted as ln, is the inverse function of the exponential function with base e. It answers the question: to what power must e be raised to get a certain number? For example, ln(e^x) = x.
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2:51
The Natural Log
Exponential Function with Base e
The exponential function e^x involves the constant e (approximately 2.718), raised to the power x. It is a fundamental function in algebra and calculus, often used to model growth or decay processes.
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4:47The Number e
Inverse Properties of Logarithms and Exponentials
Logarithms and exponentials are inverse operations, meaning ln(e^x) = x and e^(ln x) = x for x > 0. This property allows simplification of expressions involving ln and e without a calculator.
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7:30Logarithms Introduction
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